$n$-th partial sum -- generating functions

Question

Consider a finite series $S_n:=\sum_{i=0}^n i\cdot2^i$. The generating function for infinite formal series $\sum i\cdot x^i$ is $$a(x)=x\left(\frac{1}{1-x}\right)'=\frac{x}{(1-x)^2},$$ hence $$\frac{a(2x)}{1-x}=\frac{2x}{(1-x)(1-2x)^2}$$ is generating function for $\sum_{n\ge0}S_nx^n$ and $$S_n=\frac{1}{n!}\cdot\frac{d^n}{dx^n}\left.\frac{2x}{(1-x)(1-2x)^2}\right|_{x=0}$$ But this does not really satisfy me. How can I derive the closed form $S_n=(n-1)2^{n+1}+2$ (using machinery of generating functions)?